Evaluate $integral from 1 to 2 of straight x space dx$ as the limit of a sum.

Solution

Comparing $integral from 1 to 2 of straight x space dx space space with space integral from straight a to straight b of straight f left parenthesis straight x right parenthesis space dx comma space space we space get comma$
$straight f left parenthesis straight x right parenthesis space equals space straight x comma space space space straight a space equals space 1 comma space space straight b space equals space 2$
$therefore space space space straight f left parenthesis straight a right parenthesis space equals space straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight a plus straight h comma space space straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight a plus 2 straight h comma space...... straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight a plus left parenthesis straight n minus 1 right parenthesis straight h$
Now    $integral from straight a to straight b of straight f left parenthesis straight x right parenthesis space equals space stack Lt. straight h with straight h rightwards arrow 0 below left square bracket space straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis space plus space....... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
where n h = b - a
$rightwards double arrow$    $integral from straight a to straight b of xdx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight a plus straight a left parenthesis straight a plus straight h right parenthesis plus left parenthesis straight a plus 2 space straight h right parenthesis space plus space... plus left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket$
   $equals stack Lt space straight h with straight h rightwards arrow 0 below left square bracket left parenthesis straight a plus straight a plus straight a plus... to space straight n space terms right parenthesis space plus space straight h open curly brackets 1 plus 2 plus 3 plus.... plus left parenthesis straight n minus 1 right parenthesis close curly brackets right square bracket$
$equals space Lt with straight h rightwards arrow 0 below straight h open square brackets na plus straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n minus 1 plus 1 right parenthesis over denominator 2 end fraction close square brackets space space space space space open square brackets therefore 1 plus 2 plus 3 plus... plus straight n space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets equals Lt with straight h rightwards arrow 0 below straight h open square brackets straight n space straight a plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction straight h close square brackets space space equals Lt with straight h rightwards arrow 0 below open square brackets straight a space left parenthesis nh right parenthesis space plus space fraction numerator straight n space straight h left parenthesis nh minus straight h right parenthesis over denominator 2 end fraction close square brackets equals Lt with straight h rightwards arrow 0 below open square brackets straight a space left parenthesis straight b minus straight a right parenthesis space plus space fraction numerator left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b minus straight a minus straight h right parenthesis over denominator 2 end fraction close square brackets equals space straight a left parenthesis straight b minus straight a right parenthesis space plus space fraction numerator left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b minus straight a minus 0 right parenthesis over denominator 2 end fraction space equals space straight a space left parenthesis straight b minus straight a right parenthesis space plus fraction numerator left parenthesis straight b minus straight a right parenthesis squared over denominator 2 end fraction equals space left parenthesis straight b minus straight a right parenthesis open square brackets straight a plus fraction numerator straight b minus straight a over denominator 2 end fraction close square brackets space equals space left parenthesis straight b minus straight a right parenthesis open parentheses fraction numerator straight b plus straight a over denominator 2 end fraction close parentheses$
$therefore space space space space space integral from straight a to straight b of xdx space equals space 1 half left parenthesis straight b squared minus straight a squared right parenthesis$
Put a = 1, b = 2
$therefore space space space space space space space space space space space space space integral from 1 to 2 of xdx space equals space 1 half left parenthesis 4 minus 1 right parenthesis space equals space 3 over 2$

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Integrals

Evaluate $integral from 1 to 2 of straight x space dx$ as the limit of a sum.

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